jueves, 11 de abril de 2019

Find QR Decomposition (Householder Method) online step by step

Find QR Decomposition (Householder Method) online step by step

1 -1 4
1 4 -2
1 4 2
1 -1 0

https://atozmath.com/MatrixEv.aspx?q=qrdecomphh&q1=1%2C-1%2C4%3B1%2C4%2C-2%3B1%2C4%2C2%3B1%2C-1%2C0%60qrdecomphh%60D#tblSolution


Find QR Decomposition (Householder Method) ... 
 

1 -1 4
1 4 -2
1 4 2
1 -1 0
Solution:
Here A = 

1-14

14-2

142

1-10


A1 = 

1-14

14-2

142

1-10


a1 = 

1

1

1

1


||a1||=√12+12+12+12=√4=2

v1=a1-sign(a11)||a1||e1 = 

1

1

1

1
 - 2 × 

1

0

0

0
 = 

-1

1

1

1


H1=I-2⋅v1⋅vT1vT1⋅v1 = 

1000

0100

0010

0001
-24⋅

-1

1

1

1
⋅
[-1111]
 = 

0.50.50.50.5

0.50.5-0.5-0.5

0.5-0.50.5-0.5

0.5-0.5-0.50.5


H1⋅A1 = 

0.50.50.50.5

0.50.5-0.5-0.5

0.5-0.50.5-0.5

0.5-0.5-0.50.5
 × 

1-14

14-2

142

1-10
 = 

232

000

004

0-52



Now removing 1st row and 1st column, we get
A2 = 

00

04

-52


a2 = 

0

0

-5


||a2||=√02+02+(-5)2=√25=5

v2=a1-sign(a11)||a1||e1 = 

0

0

-5
 - 5 × 

1

0

0
 = 

-5

0

-5


H2=I-2⋅v1⋅vT1vT1⋅v1 = 

100

010

001
-250⋅

-5

0

-5
⋅
[-50-5]
 = 

00-1

010

-100


H2⋅A2 = 

00-1

010

-100
 × 

00

04

-52
 = 

5-2

04

00



Now removing 1st row and 1st column, we get
A3 = 

4

0



Since, H2H1A=R

H2H1A=

1000

000-1

0010

0-100
 × 

0.50.50.50.5

0.50.5-0.5-0.5

0.5-0.50.5-0.5

0.5-0.5-0.50.5
 × 

1-14

14-2

142

1-10
 = 

232

05-2

004

000
 = R


Also A=H1H2R and A=QR, ∴Q=H1H2

Q=H1H2=

0.50.50.50.5

0.50.5-0.5-0.5

0.5-0.50.5-0.5

0.5-0.5-0.50.5
 × 

1000

000-1

0010

0-100
 = 

0.5-0.50.5-0.5

0.50.5-0.5-0.5

0.50.50.50.5

0.5-0.5-0.50.5



checking Q×R=A?

Q×R = 

0.5-0.50.5-0.5

0.50.5-0.5-0.5

0.50.50.50.5

0.5-0.5-0.50.5
 × 

232

05-2

004

000
 = 

1-14

14-2

142

1-10


and A = 

1-14

14-2

142

1-10








Solution provided by AtoZmath.com
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